Enumeration of singular hypersurfaces on arbitrary complex manifolds

TL;DR

This paper derives explicit formulas for counting hypersurfaces with specific singularities in complex manifolds, using topological methods based on Euler classes, advancing enumerative geometry techniques.

Contribution

It provides a new explicit formula for enumerating singular hypersurfaces with nodes, cusps, or tacnodes in arbitrary complex manifolds, extending previous results.

Findings

Explicit formulas for hypersurfaces with singularities

Application of Euler class in enumerative geometry

Counts for hypersurfaces passing through specified points

Abstract

In this paper we obtain an explicit formula for the number of hypersurfaces in a compact complex manifold X (passing through the right number of points), that has a simple node, a cusp or a tacnode. The hypersurfaces belong to a linear system, which is obtained by considering a holomorphic line bundle L over X. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.